Problems in Quantum Mechanics and Field Theory with Mathematical Modelling

✍ Scribed by Aleksander V. Chichurin, Elena M. Ovsiyuk, Viktor M. Red’kov

• Publisher: CRC Press
• Year: 2025
• Tongue: English
• Leaves: 374
• Category: Library

✦ Table of Contents

Cover
Half Title
Title Page
Copyright Page
Contents
Foreword
Preface
Author
1. Cox scalar particle in the magnetic field in the Lobachevsky space
1.1. Introduction
1.2. The usual scalar particle in Lobachevsky space
1.3. Particle in the Lobachevsky space
1.4. Numerical study of the tunnelling effect
1.5. Conclusion
1.6. Figures
Bibliography
2. Cox scalar particle in magnetic field, the spherical space
2.1. The Cox equation for a scalar particle
2.2. Separation of the variables
2.3. Ordinary particle in Riemann space
2.4. Cox particle, analysis in the variable x = tan z
2.5. Analysis in the variable cos2 z = Z
2.6. Analysis in the variable sin2 z = x
2.7. Figures
Bibliography
3. Cox particle in the Coulomb field
3.1. Setting the problem
3.2. Separating the variables
3.3. States with zero angular momentum, l = 0
3.4. Studying the states with l = 1, 2, 3, ...
3.5. Qualitative study
3.6. Dimensionless quantities, qualitative study
3.7. Quantisation of energy, the case of minimal l = 0
3.8. Quantisation of energy at l = 1, 2, 3, ...
3.9. Nonrelativistic approximation
3.10. Frobenius solutions at l = 1, 2, ...
3.11. Qualitative study of the differential equation
3.12. Energy quantisation, non-relativistic case
3.13. Relativistic problem: polynomial solutions?
3.14. Figures
Bibliography
4. Tunnelling Dirac particles through Schwarzschild barrier
4.1. Basic facts
4.2. Analytical study of Frobenius solutions
4.3. Numerical study
4.4. Tunnelling process
4.5. Conclusions
4.6. Figures
Bibliography
5. On Maxwell equations in Schwarzschild space-time
5.1. Introduction
5.2. Separating the variables, Wigner functions
5.3. Duffin–Kemmer formalism
5.4. Relation between two formalisms
5.5. Studying equations for states with P = (–1)j
5.6. Studying the case P = (–1)j+1
5.7. The gauge degrees of freedom
5.8. Conclusions
Bibliography
6. Particle with polarisability in the Coulomb field
6.1. Introduction: starting equation
6.2. Formal exact solutions
6.3. Zero polarisability, numerical simulation
6.4. Numerical study at nonzero polarisability
6.5. Conclusions
6.6. Figures
Bibliography
7. Dirac particle in the Coulomb field in curved models
7.1. Introduction
7.2. Hydrogen atom in the Lobachevsky space
7.3. Numerical study
7.4. Hydrogen atom is spherical Riemann space
7.5. Numerical study
7.6. Solutions in the half-spaces, r ∈ (0, π/2) and r ∈ (π/2, π)
7.7. Conclusion
7.8. Figures for the problem in Lobachevsky space
7.9. Figures for the problem in spherical space
Bibliography
8. Particle with spin 1 in the Coulomb field
8.1. Separation of the variables
8.2. The case of minimal value j = 0
8.3. Nonrelativistic approximation, energy spectra
8.4. The Lorentz condition in presence of the Coulomb field
8.5. The study of equation for φ1
8.6. Equation for function φ2
8.7. Studying the first equation for Φ0
8.8. Second equation for function Φ0
8.9. Fourth-order equations, the first method
8.10. Fourth-order equations, the second method
8.11. Further study of six equations
8.12. The 4th-order differential equations
8.13. Conclusion
8.14. Figures
Bibliography
9. Geometrical modelling of the media in electrodynamics
9.1. Geometry and modelling the constitutive relations
9.2. Spinor form of Maxwell equations
9.3. Separating the variables in de Sitter models
9.4. Solutions in Minkowski space
9.5. Solutions in de Sitter space
9.6. Solutions in anti de Sitter space
9.7. Maxwell equations in Schwarzschild metric
9.8. Solutions in spherical Riemann space
9.9. Solutions in Lobachevsky space
9.10. Cylindric solutions in spherical space
9.11. Conclusions
Bibliography
10. P-asymmetric equation for a spin 1/2 particle in external fields
10.1. Gel'fand–Yaglom basis
10.2. Modified Gel'fand–Yaglom basis
10.3. On Lagrangian formulation of the model
10.4. Spinor form of the wave equation
10.5. Equations in spin-tensor form
10.6. On reducing the system to minimal form
10.7. The presence of electromagnetic field
10.8. Extension of the model to General relativity
10.9. P-noninvariant particle in the Coulomb field
10.10. Conclusions
Bibliography
11. Fermion with two mass parameters in the Coulomb field
11.1. Introduction
11.2. Separating the variables
11.3. Derivation of the 4th-order equations
11.4. Nonrelativistic approximation
11.5. Solutions of the 4th-order equations
11.6. Solutions in relativistic case
11.7. Conclusions
11.8. Figures
Bibliography
12. On modelling neutrinos oscillations by geometry methods
12.1. Fermion with three mass parameters
12.2. Reformulation of the initial equations
12.3. Characteristic equation, possible values of masses
12.4. On solutions of the characteristic equation
12.5. Diagonalization in the case of a free particle
12.6. Presence of electromagnetic fields
12.7. Quasidiagonal form of the system
12.8. Mixing the components in the system
12.9. Extension to general relativity
12.10. Model example
12.11. Plane wave solutions of Majorana type
12.12. Conclusions
Bibliography
13. Helicity operator for a spin 2 particle in magnetic field
13.1. Introduction
13.2. Helicity operator in cylindric basis, separating the variables
13.3. The system I, the free particle
13.4. The case I, particle in magnetic field
13.5. The case II, free particle
13.6. The case II, the presence of magnetic field
13.7. Numerical study of the 5th-order equation
13.8. Conclusions
Bibliography
Index